As seen in the next two propositions, for a local field \(k\) , the geometric properties of the group \(\)  interact nicely with the topological \(k\) -point structure on \(G\) . Namely, under a \(k\) -morphism of algebraic groups the image of the \(k\) -points is topologically closed, and geometrically (Zariski) dense.
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Let \(\)  be a connected \(k\)  group, then \((k)\)  is Zariski dense in \(\) .



Let \(\) , \(\)  be \(k\) -algebraic groups, and \(f\colon \rightarrow \)  be a \(k\) -morphism. Then the image of the \(k\) -points, \(f(G)\) , is a closed subgroup of \(H\) .
Moreover, the kernel of \(f\) , \(\ker (f)\)  is a \(k\) -algebraic group such that as \(k\) -analytic topological spaces \((k)/ {\ker (f)}(k)\cong f( G)\) .
